📈 Limits and Continuity

Master the foundational concepts of limits, continuity, and discontinuity essential for calculus. **What you'll learn:** - Evaluate limits using algebraic and graphical methods - Identify and classify points of discontinuity - Apply the Intermediate Value Theorem - Determine continuity at a point using the formal definition - Analyze limits at infinity and vertical asymptotes

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Question 1
What is $\lim_{x \to 2} (3x + 1)$ ?
Question 2
A function can have a limit at a point where it is not defined.
Question 3
If $\lim_{x \to 3^-} f(x) = 5$ and $\lim_{x \to 3^+} f(x) = 5$ , then $\lim_{x \to 3} f(x) =$ ___
Question 4
Which function has a removable discontinuity at $x = 2$ ?
Question 5
If $f$ is continuous on $[a, b]$ and $f(a) = -3$ and $f(b) = 7$ , then there exists a value $c$ in $(a, b)$ where $f(c) = 0$ .
Question 6
What is $\lim_{x \to \infty} \frac{3x^2 - 2x + 1}{5x^2 + 4}$ ?
Question 7
A function $f$ is continuous at $x = a$ if: (1) $f(a)$ is defined, (2) $\lim_{x \to a} f(x)$ exists, and (3) $\lim_{x \to a} f(x) =$ ___
Question 8
Which statement correctly describes the behavior of $f(x) = \frac{1}{(x - 3)^2}$ near $x = 3$ ?
Question 9
The limit $\lim_{x \to 0} \frac{\sin(x)}{x}$ exists and equals $0$ .
Question 10
Arrange the following steps in the correct order to determine whether $f(x) = \frac{x^2 - 9}{x - 3}$ is continuous at $x = 3$ :